Is there a convention on the order of multiplication of the weights with the inputs in neural nets?
The conventions I have seen tend to post-multiply rather than pre-multiply, although there are examples in the literature which adopt the opposite convention.
Some examples include:

In Deep Learning: An Introduction for Applied Mathematicians, a layer with input $x \in \mathbb R^n$ and output $f(x) \in \mathbb R^m$ is computed by
$$ f(x) = \sigma(Wx + b)$$
for a matrix $W \in \mathbb R^{m \times n}$ of weights, a vector $b \in \mathbb R^m$ of biases and an activation function $\sigma \colon \mathbb R^m \to \mathbb R^m$.

In Deep Learning by Goodfellow, Bengio and Courville, they compute a layer in chapter 6 (see p. 171) using


$$ f(x) = \max \{ 0, W^T x + c \},$$
where the $\max$ function is applied componentwise and gives the ReLU activation.

In the paper Federated Learning with Matched Averaging, the authors describe a fully-connected layer using pre-multiplication,

$$f(x) = \sigma(xW),$$
where they omit the bias to simplify notation and as before apply $\sigma$ entrywise. In this case of course we need to view $x$ as a row vector and will obtain a row vector as the result.
Ultimately, as long as you are clear it doesn't matter a great deal: in the end, all that changes is the shape and values of the matrix.